Hyperpyth: Difference between revisions
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== Hyperpyth == | == Hyperpyth == | ||
Using the fifth harmonic ([[5/1]], pentave) as an interval of equivalence, instead of the more common octave or even [[tritave]], the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a [[ | Using the fifth harmonic ([[5/1]], pentave) as an interval of equivalence, instead of the more common octave or even [[tritave]], the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a [[Macrodiatonic and microdiatonic scales|macrodiatonic]] tuning. In this case, the most suitable scales are such as would have a sharp fifth, in octaves known as "superpythagorean", so I dub this "hyperpyth". | ||
The quintessential comma of which is 28561/28125, wherein (13 the "perfect fifth")^4 = 9 (the "major third") and 5's are fungible. 13^3 (ie. a "major sixth") can also constitute the 17/5 interval, and 21/5 is liable to be an augmented sixth, (and a bonus, 19/5 can be found as a dominant seventh). This is eerily similar to the case in meantone (especially if you call the sixth a 13/8). [http://x31eq.com/cgi-bin/rt.cgi?ets=c22_c5&limit=5_9_13] | The quintessential comma of which is 28561/28125, wherein (13 the "perfect fifth")^4 = 9 (the "major third") and 5's are fungible. 13^3 (ie. a "major sixth") can also constitute the 17/5 interval, and 21/5 is liable to be an augmented sixth, (and a bonus, 19/5 can be found as a dominant seventh). This is eerily similar to the case in meantone (especially if you call the sixth a 13/8). [http://x31eq.com/cgi-bin/rt.cgi?ets=c22_c5&limit=5_9_13] | ||
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== Hyperreich? == | == Hyperreich? == | ||
{{main|Juggernaut}} | |||
Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from 5ED5, 10ED5, 15ED5) and 596-cent (from [[14ed5|14ED5]], which is a slightly compressed [[6edo|6EDO]]) intervals are the closest approximations. That is, until [[19ed5|19ED5]] (14+5) which is a very slightly stretched [[13edt|13EDT]] (Bohlen-Pierce) scale, and [[24ed5|24ED5]] which is something completely different. | Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from 5ED5, 10ED5, 15ED5) and 596-cent (from [[14ed5|14ED5]], which is a slightly compressed [[6edo|6EDO]]) intervals are the closest approximations. That is, until [[19ed5|19ED5]] (14+5) which is a very slightly stretched [[13edt|13EDT]] (Bohlen-Pierce) scale, and [[24ed5|24ED5]] which is something completely different. | ||
[[Category:Hyperpyth| ]] | [[Category:Hyperpyth| ]] <!-- main article --> | ||
[[Category: | [[Category:Rank-2 temperaments]] | ||
[[Category: | [[Category:Non-octave temperaments]] | ||